Radio Network Distributed Algorithms in the Unknown Neighborhood Model

The paper deals with radio network distributed algorithms where nodes are not aware of their one hop neighborhood. Given an n-node graph modeling a multihop network of radio devices, we give a O(log^2 n) time distributed algorithm that computes w.h.p., a constant approximation value of the degree of each node. We also provide a O( \Delta log n + log^2 n) time distributed algorithm that computes w.h.p., a constant approximation value of the local maximum degree of each node, where the global maximum degree \Delta of the graph is not known. Using our algorithms as a plug-and-play procedure, we show that many existing distributed algorithms requiring the knowledge of to execute efficiently can be run with essentially the same time complexity by using the local maximum degree instead of . In other words, using the local maximum degree is sufficient to break the symmetry in a local and efficient manner. We illustrate this claim by investigating the complexity of some basic problems. First, we investigate the generic problem of simulating any classical message passing algorithm in the radio network model. Then, we study the fundamental edge/node coloring problem in the special case of unit disk graphs. The obtained results show that knowing the local maximum degree allows to coordinate the nodes locally and avoid interferences in radio networks

On the Complexity of Distributed Splitting Problems

One of the fundamental open problems in the area of distributed graph algorithms is the question of whether randomization is needed for efficient symmetry breaking. While there are fast, $\text{poly}\log n$-time randomized distributed algorithms for all of the classic symmetry breaking problems, for many of them, the best deterministic algorithms are almost exponentially slower. The following basic local splitting problem, which is known as the \emph{weak splitting} problem takes a central role in this context: Each node of a graph $G=(V,E)$ has to be colored red or blue such that each node of sufficiently large degree has at least one node of each color among its neighbors. Ghaffari, Kuhn, and Maus [STOC '17] showed that this seemingly simple problem is complete w.r.t. the above fundamental open question in the following sense: If there is an efficient $\text{poly}\log n$-time determinstic distributed algorithm for weak splitting, then there is such an algorithm for all locally checkable graph problems for which an efficient randomized algorithm exists. In this paper, we investigate the distributed complexity of weak splitting and some closely related problems. E.g., we obtain efficient algorithms for special cases of weak splitting, where the graph is nearly regular. In particular, we show that if $\delta$ and $\Delta$ are the minimum and maximum degrees of $G$ and if $\delta=\Omega(\log n)$, weak splitting can be solved deterministically in time $O\big(\frac{\Delta}{\delta}\cdot\text{poly}(\log n)\big)$. Further, if $\delta = \Omega(\log\log n)$ and $\Delta\leq 2^{\varepsilon\delta}$, there is a randomized algorithm with time complexity $O\big(\frac{\Delta}{\delta}\cdot\text{poly}(\log\log n)\big)$

How Many Pairwise Preferences Do We Need to Rank A Graph Consistently?

We consider the problem of optimal recovery of true ranking of $n$ items from a randomly chosen subset of their pairwise preferences. It is well known that without any further assumption, one requires a sample size of $\Omega(n^2)$ for the purpose. We analyze the problem with an additional structure of relational graph $G([n],E)$ over the $n$ items added with an assumption of \emph{locality}: Neighboring items are similar in their rankings. Noting the preferential nature of the data, we choose to embed not the graph, but, its \emph{strong product} to capture the pairwise node relationships. Furthermore, unlike existing literature that uses Laplacian embedding for graph based learning problems, we use a richer class of graph embeddings---\emph{orthonormal representations}---that includes (normalized) Laplacian as its special case. Our proposed algorithm, {\it Pref-Rank}, predicts the underlying ranking using an SVM based approach over the chosen embedding of the product graph, and is the first to provide \emph{statistical consistency} on two ranking losses: \emph{Kendall's tau} and \emph{Spearman's footrule}, with a required sample complexity of $O(n^2 \chi(\bar{G}))^{\frac{2}{3}}$ pairs, $\chi(\bar{G})$ being the \emph{chromatic number} of the complement graph $\bar{G}$. Clearly, our sample complexity is smaller for dense graphs, with $\chi(\bar G)$ characterizing the degree of node connectivity, which is also intuitive due to the locality assumption e.g. $O(n^\frac{4}{3})$ for union of $k$-cliques, or $O(n^\frac{5}{3})$ for random and power law graphs etc.---a quantity much smaller than the fundamental limit of $\Omega(n^2)$ for large $n$. This, for the first time, relates ranking complexity to structural properties of the graph. We also report experimental evaluations on different synthetic and real datasets, where our algorithm is shown to outperform the state-of-the-art methods.Comment: In Thirty-Third AAAI Conference on Artificial Intelligence, 201